Hall Algebras And Lie Algebras Associated To Cyclic Complexes

Let A be a finite dimensional algebra of finite global dimension over a finite field k = Fq.In 2011,Bridgeland considered the Hall algebra of the category of 2-cyclic complexes over projective A-modules,he defined an algebra DH2red(A)by taking some localization and reduction,which is called Bridgeland's Hall algebra of A.In particular,if A is hereditary,this algebra provides a realization of the entire quantum group associated to A.Let m be a fixed non-negative integer,inspired by the work of Bridgeland,Chen and Deng considred the Hall algebra of the category Cm((?)),which consists of m-cyclic com-plexes over projective A-modules.In this thesis,we mainly study the Hall algebras and Lie algebras associated to the category Cm((?)).Our main results consist of the following three parts:(1)Let A be a hereditary algebra and B a tilted algebra of A.We consider Bridge-land's Hall algebra DH2red(B)of B,and obtain an algebra homomorphism from the quan-tum group associated to A to Bridgeland's Hall algebra of B.In particular,for the APR-tilting,this homomorphism provides a realization of Lusztig's symmetries of quantum groups via Bridgeland's Hall algebras.(2)Let A be a hereditary algebra and m a fixed integer such that m = 0 or m>2.We first introduce the m-periodic lattice algebra of A,and then prove that it is isomorphic to Bridgeland's Hall algebra DHm(A)of Cm((?)).Moreover,we show that there exist certain Heisenberg double structures hidden in DHm(A).(3)Let A = kQ be the path algebra of a Dynkin quiver Q.We first prove the exis-tence of Hall polynomials in C1((?)),and then establish a relationship bewteen the Hall numbers for indecomposable objects in C1((?))and those for A-modules.Using the com-mutators of degenerate Hall multiplication,we define a Lie algebra n+,which is spanned by the isomorphism classes of non-acyclic indecomposable objects in C1((?)).Then we characterize the Lie algebra n+ by generators and relations.If Q is bipartite,i.e.,each vertex of Q is either a sink or a source,n+ is isomorphic to the positive part of the cor-responding semisimple complex Lie algebra.If Q is the linearly oriented quiver of type An,n+ is isomorphic to the free 2-step nilpotent Lie algebra with n-generators.In other cases,they provide variant nilpotent Lie algebras.Furthermore,we give a description of the root system of n+.When Q is of type A,we prove the achieved relations are exactly the generating relations of n+.